The O'Brien family was the family of Miles and Keiko O'Brien.
The two were married on October 24, 2367, after Data introduced them aboard the USS Enterprise-D. (TNG: "Data's Day")
Their first child, Molly, was born in 2368. (TNG: "Disaster")
Miles and Keiko went to Earth in 2369 to celebrate the 100th birthday of Miles' mother. (DS9: "Dax")
After the O'Briens moved to Deep Space 9, their second child and first son, Kirayoshi, was born in early 2373 after being carried to term by Kira Nerys. (DS9: "Body Parts", "The Begotten")